Z (Normal Distribution) Tests

For large (50 or more observations) normally distributed samples, normal distribution tests are equivalent to Student t tests.

Normal data

You may either compare the means of two independent random samples or compare the mean of one sample with a known population mean. Note that for large degrees of freedom, Student's t distribution is approximately normal (Altman, 1991; Armitage and Berry, 1994).

See the examples for t tests and consider these in the context of larger samples.

You will gain a little more sensitivity by using a normal distribution test instead of its equivalent Student t test but you must have good reason to believe that your data have been drawn from a normal distribution. Student t tests are less sensitive than normal distribution tests to small deviations from normality; use t tests if you have any doubt. If your data are clearly non-normal then you should consider using a nonparametric alternative such as the Wilcoxon signed ranks test or the Mann-Whitney U test.

The single sample test statistic is calculated as:

- where x bar is the sample mean, s² is the sample variance, n is the sample size, µ is the specified population mean and z is a quantile from the standard normal distribution.

The unpaired test statistic is calculated as:

- where x-bar 1 and x-bar 2 are the sample means, s1² and s2² are the sample variances, n1 and n2 are the sample sizes and z is a quantile from the standard normal distribution.

Log-normal data

For samples from a log-normal distribution (logs from a normal distribution) you may wish to construct an interval analogous to the confidence interval for the mean of a sample from a normal distribution. StatsDirect gives you the geometric mean (arithmetic mean of logs) and a reference range as:

- where g is geometric mean, ln is natural logarithm, n is sample size and z is a quantile from the standard normal distribution (alpha/2 quantile for a 100*(1-alpha)% confidence interval).